Half empty, half full and the possibility of agreeing to
disagree
Alexander Zimper
1Working Paper Number 58
1
Half empty, half full and the possibility of agreeing
to disagree
Alexander Zimper
yOctober 31, 2007
Abstract
Aumann (1976) derives his famous we cannot agree to disagree result under the assumption of rational Bayesian learning. Motivated by psychological evi-dence against this assumption, we develop formal models of optimistically, resp. pessimistically, biased Bayesian learning within the framework of Choquet ex-pected utility theory. As a key feature of our approach the posterior subjective beliefs do, in general, not converge to “true” probabilities. Moreover, the posteri-ors of di¤erent people can converge to di¤erent beliefs even if these people receive the same information. As our main contribution we show that people may well agree to disagree if their Bayesian learning is psychologically biased in our sense. Remarkably, this …nding holds regardless of whether people with identical priors apply the same psychologically biased Bayesian learning rule or not. A simple example about the possibility of ex-post trading in a …nancial asset illustrates our formal …ndings. Finally, our analysis settles a discussion in the no-trade literature (cf. Dow, Madrigal, and Werlang 1990, Halevy 1998) in that it clari…es that ex-post trade between agents with common priors and identical learning rules is only possible under asymmetric information.
Keywords: Common Knowledge, No-Trade Results, Rational Bayesian Learn-ing, Bounded Rationality, Choquet Expected Utility Theory, Bayesian UpdatLearn-ing, Dynamic Inconsistency
JEL Classi…cation Numbers: D81, G20.
I would like to thank Alexander Ludwig and Elias Khalil for helpful comments and suggestions.
1
Introduction
1.1
Motivation
Aumann (1976) proves that “If two people have the same priors, and their posteriors for an event A are common knowledge, then these posteriors are equal”(p. 1236). This celebrated we cannot agree to disagree result has been derived under the assumption that people’s posterior beliefs result from rational Bayesian learning. However, several studies in the psychological literature show that real-life agents systematically violate this assumption in that their learning behavior is prone to e¤ects such as “myside bias” or “irrational belief persistence”(cf., e.g., the references in chapter 9, Baron 2007). For example, in an early contribution to this literature, Lord, Ross, and Lepper (1979) con-duct an experiment in which agents’posteriors diverge despite the fact that all agents have received the same information.1 Moreover, de…nitions of several psychological phe-nomenons such as delusions, depressions etc. are based on the observation that di¤erent subjects may interpret identical information in di¤erent ways (cf. Beck 1976).
The contributions of this paper are two-fold. At …rst, we introduce in this paper a formal model of Bayesian learning with a myside bias. More precisely, we introduce the notion of optimistically, resp. pessimistically, biased Bayesian learning. With these de…nitions we formally describe the di¤erence between “half empty” versus “half full” attitudes in the context of interpreting new information. In contrast to the standard model of rational Bayesian learning (e.g., Tonks 1983, Viscusi and O’Connor 1984, Vis-cusi 1985), the posterior beliefs of biased Bayesian learning do not converge to the “true” probabilities. Furthermore, the posterior beliefs of di¤erent agents may well converge to di¤erent beliefs even if these agents always receive the same information. As our second contribution, we demonstrate that people may agree to disagree if their Bayesian learning is psychologically biased. This result holds regardless of whether people apply the same Bayesian learning rule or not.
Key to our analysis is the assumption that people are Choquet expected utility (CEU) rather than expected utility (EU) decision makers. CEU theory (Schmeidler 1989, Gilboa 1987) is a generalization of EU theory that admits for the integration of a vNM function with respect to non-additive probability measures (capacities). Properties of such capacities are used for the formal description of ambiguity attitudes which may explain Ellsberg (1961) paradoxes. Ellsberg paradoxes demonstrate systematic violations 1The subjects in this experiment were confronted with two purported statistical studies, one study
supporting the other study rejecting the claim that capital punishment has a crime deterrence e¤ect. For analogous results in the context of Bayesian updating of subjective probabilities see Pitz, Downing, and Reinhold (1967) and Pitz (1969).
of Savage’s (1954) “sure thing principle”. The sure thing principle, however, ensures that there is a unique way of deriving ex-post preferences from ex-ante preferences, implying a unique Bayesian update rule for the additive probabilities of subjective EU theory. The picture is di¤erent for the non-additive probability measures of CEU theory for which several perceivable Bayesian update rules exist (cf. Gilboa and Schmeidler 1993, Eichberger, Grant and Kelsey 2006, Siniscalchi 2001, 2006). Following Gilboa and Schmeidler’s (1993) psychological interpretation we consider the extreme cases of the optimistic, resp. pessimistic, update rule, which we apply to non-additive probability measures de…ned as neo-additive capacities in the sense of Chateauneuf, Eichberger and Grant (2006). Our resulting de…nition of optimistically, resp. pessimistically, biased agents combines the standard model of rational Bayesian learning with an optimistic, resp. pessimistic, attitude towards the interpretation of new information.
We present two di¤erent results of the type that people may agree to disagree if their learning rules are psychologically biased. Our …rst result (proposition 1) shows that if two people have the same prior, apply di¤erent learning rules, and their posteriors for an
event A =2 f?; g are common knowledge, then these posteriors will be di¤erent even in
case they have identical information partitions. Within the appropriate framework this result is easily derived. However, beyond the mere formal result our …nding addresses an important behavioral issue. Aumann (1976) writes “In private conversation, Tversky has suggested that people may also be biased because of psychological factors, that may make them disregard information that is unpleasant or does not conform to previously formed notions” (p. 1238). There is no way of describing such psychological biases of real-life people within Aumann’s framework. Within our approach, however, the resulting “myside bias”has a straightforward interpretation as people’s di¤erent attitudes towards the interpretation of information due to psychological predispositions such as the “half-empty glass” versus the “half-full glass” attitude.
Whereas our …rst result applies to people who use di¤erent rules of Bayesian learning, our second result (proposition 2) refers to the case of identical learning rules. We …nd that if two people have the same prior, apply the same learning rule, and their posteriors
for an event A =2 f?; g are common knowledge, then these posteriors can be di¤erent
in case they have di¤erent information partitions. Thus, neither in the case where people have the same information partitions nor in the case where people apply the same update rule does Aumann’s conclusion obtain when Bayesian learning is psychologically biased in our sense. To the contrary, according to our results a di¤erence in posteriors that are common knowledge is the rule rather than the exception when people are psychologically biased.
1.2
Relationship to no-trade results
Combined with Harsanyi’s (1967) common priors doctrine Aumann’s we cannot agree to disagree result has been very in‡uential in information economics. Especially the so-called no-trade theorems - basically stating that there should be no ex-post trade in …nancial assets if the agents are rational - are based on Aumann’s approach (cf., e.g., Milgrom 1981, Milgrom and Stokey 1982, Samet 1990, Morris 1994, Bonanno and Nehring 1999). The connection between Aumann’s we cannot agree to disagree result and the impossibility of ex-post trade in …nancial assets is straightforward. Under the assumption that agents have di¤erent preferences for such assets if and only if they have di¤erent beliefs about the assets’future returns, there are strict incentives for ex-post trade if and only if the agents have di¤erent posterior beliefs. Since the market-price of such assets is common knowledge between the trading agents, any trade would result in
the traders’common knowledge that their posteriors must be di¤erent.2
Since no trade-results are seemingly at odds with reality, there are several contri-butions in the literature investigating the robustness of no-trade results with respect to a weakening of Aumann’s assumptions. One line of research discusses concepts of bounded rationality that weaken the rationality assumptions of Aumann’s epistemic framework. For example, information structures have been considered that are non-partitional (Bacharach 1985, Samet 1990, Geneakoplos 1992, Rubinstein and Wolinsky 1990) or concepts of “almost”common-knowledge have been introduced (Neeman 1996). In contrast to this literature our approach fully adopts Aumann’s epistemic framework. The agents of our model are boundedly rational not with respect to their logical capa-bility but with respect to their psychological bias in interpreting new information.
Closer to our own approach is a second line of research on no-trade results that consid-ers decision theoretic alternatives to EU theory. In an early contribution Dow, Madrigal and Werlang (1990) already provide an example in which ex-post trading becomes pos-sible because agents update their non-additive beliefs according to the Dempster-Shafer rule which is at the heart of our de…nition of pessimistically biased Bayesian learning. Dow et al. thereby assume asymmetric information and common non-additive priors so that their example can be regarded as an illustration of our proposition 2 for the special case of pessimistically biased agents.
Halevy (1998, 2004) claims that the …nding of Dow et al. can be extended to the case of symmetric information so that there might occur ex-post trading between agents with common priors and identical information partitions if their beliefs are non-additive. More precisely, Halevy writes:
2Note that the typical assumption of “strictly risk averse”traders (e.g., Milgrom 1981, Milgrom and
“A similar result appears in Dow et al (1990). Their result, as noted by Epstein and Le Breton (1993) and as our present example illustrates, relies merely on dynamic inconsistency of the individual agents. Their claim that trade is a result of asymmetric information is not accurate: we show below that it could be reached with completely symmetric information and even with a common prior.” (footnote 17, p. 20 in Halevy 1998)
In the light of our propositions 1 and 2, we take a somewhat di¤erent view from Halevy. Namely, Dow et al.’s conclusion is indeed accurate under their assumption of an identical update rule for all agents: our analysis demonstrates that agents with an identical update rule cannot agree to disagree if they have identical information partitions. Our own asset-trade example in section 5 of this paper therefore establishes the existence of ex-post trade between agents with symmetric information and common priors if and only if the agents have di¤erent update rules. Since Halevy’s example does not consider di¤erent update rules, his …nding is apparently at odds with our own results. As it turns out the di¤erence between Dow et al. and our conclusion, on the one hand, and Halevy’s conclusion, on the other hand, is due to di¤erent notions of common priors. Halevy’s example is based on Yaari’s (1987) dual theory in which additive probabilities are transformed into non-additive beliefs by some transformation function. Halevy speaks of common priors because his agents have common additive probabilities. But since these agents apply di¤erent transformation functions, their resulting non-additive priors are no longer identical. According to our approach and the approach of Dow et al. the assumption of common priors is therefore violated in Halevy’s example.
Rubinstein and Wolinsky (1990, Remark p. 190) argue that Milgrom and Stokey’s no-trade result applies to all decision theories under uncertainty which satisfy dynamic consistency. Halevy (2004) reports the interesting fact that there might even be ex-post trading between dynamically consistent agents if these agents violate consequentialism. While EU decision makers satisfy, by the sure-thing principle, dynamic consistency as well as consequentialism, the agents of our model only satisfy consequentialism and the possibility of agreeing to disagree exclusively results from their dynamically inconsistent preferences (cf. Epstein and Le Breton 1993, Sarin and Wakker 1998).
The subsequent analysis is structured as follows. In section 2 we describe our decision-theoretic framework which we combine in section 3 with the standard model of rational Bayesian learning. Section 4 recalls Aumann’s (1976) epistemic framework and presents our …rst agreeing to disagree result. A simple example in section 5 about the possibility
of ex-post asset trade illustrates this …rst result. Our second agreeing to disagree result is stated and proved in section 6. Section 7 concludes.
2
Preliminaries: The decision-theoretic framework
As in Aumann (1976) we consider a measurable space ( ; B) with B denoting a -algebra
on the state space . As a generalization of Aumann’s assumption of EU decision
makers, however, we consider a CEU rather than an EU decision maker.3 In contrast
to EU theory, CEU theory admits for non-additive probability measures, i.e., capacities,
whereby a capacity : B ! [0; 1] must satisfy
(i) (;) = 0, ( ) = 1
(ii) A B ) (A) (B) for all A; B 2 B:
Additional properties of capacities are used in the literature for formal de…nitions of, e.g., ambiguity and uncertainty attitudes (Schmeidler, 1989; Epstein, 1999; Ghirardato and Marinacchi, 2002), pessimism and optimism (Eichberger and Kelsey, 1999; Wakker, 2001), as well as sensitivity to changes in likelihood (Wakker, 2004).
In our model of non-rational Bayesian learning we restrict attention to a class of capacities that are de…ned as neo-additive capacities in the sense of Chateauneuf, Eich-berger, and Grant (2006). Neo-additive capacities stand for deviations from additive
probabilities such that a parameter (degree of ambiguity) measures the lack of
con…-dence the decision maker has in some subjective additive probability distribution . In
addition, a second parameter measures the degree of optimism versus pessimism by
which the decision maker resolves his ambiguity.
De…nition: A neo-additive capacity, , is de…ned, for some ; 2 [0; 1], by
(A) = ( !o(A) + (1 ) !p(A)) + (1 ) (A) (1)
3CEU theory was …rst axiomatized by Schmeidler (1986, 1989) within the Anscombe and Aumann
(1963) framework, which assumes preferences over objective probability distributions. Subsequently, Gilboa (1987) as well as Sarin and Wakker (1992) have presented CEU axiomizations within the Savage (1954) framework, assuming a purely subjective notion of likelihood. CEU theory is equivalent to cumulative prospect theory (Tversky and Kahneman, 1992; Wakker and Tversky, 1993) restricted to the domain of gains (compare Tversky and Wakker, 1995). Moreover, as a representation of preferences over lotteries, CEU theory coincides with rank dependent utility theory as introduced by Quiggin (1981, 1982).
for all A 2 B such that is some additive probability measure and we have for the non-additive capacities !o !o(A) = 1 if A 6= ; !o(A) = 0 if A = ; and !p respectively !p(A) = 0 if A 6= !p(A) = 1 if A = .
Let the state space be …nite and denote by f (!) the outcome of the Savage-act
f in state ! 2 . The Choquet expected utility of a Savage-act f with respect to a
neo-additive capacity is CEU (f; ) = max !2 u (f (!)) + (1 ) min!2 u (f (!)) +(1 ) X !2 (!) u (f (!)), (2)
with u ( ) denoting von Neumann-Morgenstern utility indices.4
In contrast to EU preferences there exist for CEU preferences several possibilities of deriving from ex ante preferences ex post preferences, i.e., preferences conditional on the fact that some event B has occurred. Following Gilboa and Schmeidler (1993) we
focus on so-called f -Bayesian update rules for preferences over Savage acts. That is,
we consider some collection of conditional preference orderings,n fBo for all events B, such that for all acts g; h
g fB h, (g; B; f; :B) (h; B; f;:B) (3)
where (g; B; f; :B) denotes the act that gives consequences g (!) for all ! 2 B and consequences f (!) for all ! 2 :B. Gilboa and Schmeidler show that CEU preferences 4Ludwig and Zimper (2006a) show that the CEU of an act with respect to a neo-additive capacity
can be equivalently described by the -maxmin expected utility with respect to multiple priors ( -MEU) of an act which encompasses the original multiple priors approach of Gilboa and Schmeidler (1989) as a special case (see, e.g., Ghirardato et al., 1998; Ghirardato et al., 2004; Siniscalchi, 2005). In particular, we have equivalence between the CEU with respect to neo-additive capacities and the -MEU with respect to so-called "-contaminated priors used in Bayesian statistics (Berger and Berliner, 1986) that may be interpreted as neo-additive capacities.
on Savage acts are updated to conditional CEU preferencesn fBo for all events B if and only if f is an act such that for some event E 2
f = (x ; E; x ;:E) ; (4)
where x denotes the best and x denotes the worst consequence possible. The di¤erent possible speci…cations of E in (4) can result in a multitude of di¤erent f -Bayesian update rules if dynamic consistency is violated. For example, for the so-called optimistic update rule f is the constant act where E = ;. That is, under the optimistic update rule the null-event becomes associated with the worst consequence possible. Gilboa and Schmeidler (1993) o¤er the following psychological motivation for this update rule:
“[...] when comparing two actions given a certain event A, the decision maker implicitly assumes that had A not occurred, the worst possible outcome [...] would have resulted. In other words, the behavior given A [...] exhibits ‘happiness’ that A has occurred; the decisions are made as if we are always in ‘the best of all possible worlds’.”
As corresponding optimistic Bayesian update rule for conditional beliefs of CEU decision makers obtains
opt(A
j B) = (A(B)\ B): (5)
For the pessimistic (=Dempster-Shafer) update rule f is the constant act where E = S, associating with the null-event the best consequence possible. Gilboa and Schmeidler:
“[...] we consider a ‘pessimistic’ decision maker, whose choices reveal the hidden as-sumption that all the impossible worlds are the best conceivable ones.”
The corresponding pessimistic Bayesian update rule for CEU decision makers is
pess
(Aj B) = (A[ :B) (:B)
1 (:B) : (6)
Observation 1: Suppose A; B =2 f?; g.
(i) An application of the optimistic update rule (5) to a prior belief (1) results in the conditional belief
opt(A
j B) = optB + 1 opt
with
opt
B = + (1 ) (B).
(ii) An application of the pessimistic update rule (6) to a prior belief (1) results in the conditional belief
pess(A j B) = (1 pessB ) (Aj B) with pess B = (1 ) (1 ) + (1 ) (B).
Proof: Relegated to the appendix.
Let A; B =2 f?; g and observe that
pess(A
j B) < opt(Aj B) , (7)
if > 0, and
pess(A
j B) < (A j B) < opt(Aj B)
if > 0, 2 (0; 1), and (A j B) 2 (0; 1). For the ex post evaluation of any Savage act
f we therefore have
CEU (f; pess(Aj B)) EU (f; (Aj B)) CEU f; opt(Aj B) ,
whereby these inequalities are strict in non-trivial cases.
3
Psychologically biased Bayesian learning
Let us …rst recall the standard model of rational Bayesian learning which obtains as a special case of our approach. Following Viscusi and O’Connor (1984) we consider an
agent who has a prior probability distribution over the parameter of a
Binomial-distribution ( being the true probability of some event A) such that this prior
distri-bution belongs to the family of Beta distridistri-butions. The agent’s subjective prior about
, denoted , is the expected value of this prior Beta distribution, i.e., (A) = + for
given parameters and . Let (Aj In) denote the agent’s posterior about derived
from rational Bayesian learning whereby the event In denotes information equivalent to
Rational Bayesian learning then results in a posterior Beta distribution about with
expected value +kn
+ +n implying for the posterior belief
(Aj In) =
+
+ + n (A) +
n
+ + n n (8)
where n denotes the sample mean knn. Since kn
n converges for n ! 1 in probability to
(A), we have, for any c > 0, lim
n!1prob (j (A j In) (A)j < c) = 1, (9)
which we abbreviate henceforth as lim
n!1 (Aj In) = (A).
That is, under the assumption of rational Bayesian learning the posterior beliefs con-verges to the true probability if the number of trials (=sample size) approaches in…nity.5
De…nition: Psychologically biased Bayesian learning.
(i) We say an agent is optimistically biased if his posterior beliefs result from
an application of the optimistic Bayesian update rule (5) to a neo-additive prior (1) such that the additive part of the posterior re‡ects rational Bayesian learning in the sense of (8).
(ii) We say an agent is pessimistically biased if his posterior beliefs result from an application of the pessimistic Bayesian update rule (6) to a neo-additive prior (1) such that the additive part of the posterior re‡ects rational Bayesian learning in the sense of (8).
Let us assume that an agent who receives information in the n-th trial, did also receive information in the proceeding trials; that is, the events In, n = 1; 2; ::, form
a nested sequence I1 I2 ::: . As a consequence the corresponding sequence of
probabilities (I1) ; (I2) ; ::: is monotonically decreasing, implying the existence of a
unique limit point, i.e.,
lim
n!1 (In) = b2 [0; (I1)]. (10)
5A similar result obtains when the agent has a normally distributed prior over the mean of some
Moreover, in the plausible case that the agent does not expect to collect new information forever we have limn!1 (In) = 0. Because of (9) and (10) we obtain for the limit beliefs
of psychologically biased Bayesian learning:
Observation 2: Suppose A =2 f?; g.
(i) If the agent is optimistically biased, then lim
n!1
opt(A
j In) = opt+ 1 opt (A)
such that
opt
2 + (1 ) (
I1)
; 1 . (ii) If the agent is pessimistically biased, then
lim n!1 pess(A j In) = (1 pess) (A) such that (1 pess)2 0; (1 ) (I1) (1 ) + (1 ) (I1) .
Observe that opt > 0if and only if > 0 and > 0. Analogously, (1 pess) < 1 if and only if > 0 and < 1.
Corollary: Suppose (A)2 (0; 1) and > 0.
(i) The posteriors of an optimistically biased agent with > 0 converge to some
belief strictly greater than the true probability (A). In particular, if lim
n!1 (In) = 0,
then the agent’s posteriors converge to extreme optimism, i.e., lim
n!1
opt(A
j In) = 1.
(ii) The posteriors of a pessimistically biased agent with < 1 converge to some belief strictly greater than the true probability (A). If
lim
n!1 (In) = 0,
then the agent’s posteriors converge to extreme pessimism, i.e., lim pess(Aj I ) = 0.
The corollary demonstrates that psychologically biased Bayesian learning in our sense violates the two standard paradigms of rational Bayesian learning. Firstly, the posterior “subjective”beliefs do not converge to the “objective”probabilities in an in…nite learning process. Secondly, the posteriors of two di¤erent agents do not converge to the same limit belief if they receive the same information but interpret it di¤erently.
4
A …rst result: Identical information partitions
Throughout this paper we adopt Aumann’s (1976) original epistemic framework. We
consider two partitions P1 and P2 of a non-empty state-space which are interpreted
as the information space of agent 1, respectively 2. Denote by Pi(!), with i 2 f1; 2g,
the member of Pi that contains ! 2 . We say that i knows event A 2 B in state !
i¤ Pi(!) A. Moreover, let P1 ^ P2 denote the …nest partition of B that is coarser
than P1 and P2 (i.e., the meet of P1 and P2). Following Aumann’s de…nition, we say
that event A 2 B is common knowledge between agent 1 and 2 in state ! i¤ P (!) A
whereby P (!) is the member of P1^ P2 containing ! 2 .
Our …rst agreeing to disagree result considers the situation in which agents have identical information partitions but apply di¤erent update rules.
Proposition 1: Consider the following assumptions:
(A1) The agents have identical neo-additive priors, i.e., 1 = 2 , such that
> 0.
(A2) The agents have identical information partitions P1 =P2 6= f g.
(A3) Agent 1 is optimistically whereas agent 2 is pessimistically biased.
(A4) The agents’ posteriors are common knowledge in some state of the world
! 2 .
Then the agents’posterior beliefs about any event A =2 f?; g are di¤erent.
Proof: Suppose that the posteriors are common-knowledge in ! 2 . By
assump-tion, agent 1 is optimistically and agent 2 is pessimistically biased, implying
1(Aj P1(! )) = opt(Aj P1(! )) 2(Aj P2(! )) = pess(Aj P2(! )).
Moreover, P1 = P2 implies P1 = P2 = P1 ^ P2 so that P (! ) = P1(! ) = P2(! ).
By inequality (7), the agents’posteriors 1(Aj P (! )) and 2(Aj P (! )) are therefore
di¤erent for every event A =2 f?; g.
Proposition 1 shows that, except for degenerate cases, optimistically and pessimisti-cally biased agents have in the ex-post situation always strict incentives to bet with each other. By the corollary to observation 2, these incentives will get stronger the more in-formation the agents collect. This result holds regardless of the fact whether the agents collect identical or di¤erent information.
While the formal proof of proposition 1 is simple, it reveals a fundamental di¤erence between Aumann’s concept of information and our approach. According to Aumann, any di¤erences in the beliefs of di¤erent agents are caused by di¤erent information received by the agents. According to our approach, di¤erences in the beliefs of agents may also result because of di¤erent psychological attitudes with respect to the interpretation of new information. That is, while one agent might have a “half-full” attitude, another agent may have a “half-empty” attitude when interpreting the same fact.
5
An illustrative example: Asset-trading
We illustrate proposition 1 by a simple example in which ex-post asset-trading happens in every state of the world due to di¤erent ex-post evaluations of the asset. Moreover, these di¤erent ex-post evaluations are common knowledge to the agents despite the fact that their ex ante evaluations and their information partitions are identical.
Assume that agent 2 owns in period 1 a …nancial asset which gives vNM utility of 1 in case an investment project is successful and an utility of 0 in case it is not. Before it will be revealed in period 3 whether the project is successful or not, there will be news about the project’s progress, either good or bad, in period 2. Let the relevant state space be given as
=fSG; SB; F G; F Bg
whereby the event G = fSG; F Gg stands for good and the event B = fSB; F Bg stands for bad news in period 2. Accordingly, S = fSG; SBg is the event of success and F = fF G; F Bg is the event of failure. The information partitions P1(t) ;P2(t) , t 2 f1; 2g,
in period t = 1 are
P1(1) =P2(1) =f g .
CEU1(f; ) = CEU2(f; ) = + (1 ) (S).
As a consequence, there is no strict incentive for the agents to trade the asset in the ex-ante situation.
Consider now the following information partitions at period 2 P1(2) =P2(2) =ffSG; F Gg ; fSB; F Bgg
and assume that agent 1 applies optimistically and agent 2 applies pessimistically biased Bayesian learning upon learning the news x 2 fG; Bg. Agent 1, resp. 2, then evaluate holding the asset in the ex-post situation as
CEU1 f; opt( j x) = opt(S j x) ,
resp.
CEU2(f; pess( j x)) = pess(Sj x) .
By (7), we have for ; 2 (0; 1)
CEU2(f; pess( j x)) < (S j x) < CEU1 f; opt( j x) .
Thus, regardless of whether the news turn out good or bad agent 1 ex-post evaluates the asset strictly higher than agent 2. As a consequence there will be ex-post trade
in the asset in every state of the world. For example, at price (S j x) agent 2 would
strictly prefer to sell the asset while agent 1 would strictly prefer to buy it.
Remark. Observe that if both agents were EU decision makers, i.e., = 0, there
would be no strict incentive for ex-post trading if there is no strict incentive for ex-ante trading. As this example shows, this is not necessarily true for CEU decision makers
because of the possibility of dynamically inconsistent CEU preferences.6 According to
our concept of psychologically biased Bayesian learning, the incentive for ex-post trading results in the example from the agents’di¤erent psychological attitudes with respect to the interpretation of new information.
6Ludwig and Zimper (2006b) demonstrate that sophisticated (Strotz 1956, Pollak 1968) CEU decision
makers may have even stronger incentives for intrapersonal commitment than sophisticated hyperbolic discounting decision makers in the sense of Laibson (1997) and Frederick, Loewenstein, and O’Donoghue (2002).
6
A second result: Identical learning rules
Our second agreeing to disagree result applies to people who use the same psychologically biased learning rule but have di¤erent information partitions.
Proposition 2: Consider the following assumptions:
(A1’) The agents have identical neo-additive priors, i.e., 1 = 2 , such that
> 1.
(A2’) Both agents are either optimistically or pessimistically biased.
(A3’) The agents’ posteriors are common-knowledge in some state of the world
! 2 .
(A4’) The agents’priors satisfy (P1(! ))6= (P2(! )) whereby P1(! ) ; P2(! )6=
.
Then the agents’posterior beliefs about any event A =2 f?; g are di¤erent.
Observe that assumption (A4’), i.e., (P1(! )) 6= (P2(! )), cannot hold if the
agents have identical priors and identical information partitions. That is, the result of proposition 2 only applies in situations of asymmetric information, i.e., P1 6= P2, such
that the two events P1(! ) and P2(! ) are not equally likely according to the agents’
common prior.
Before we turn to the proof of proposition 2 consider the following example which illustrates the intuition behind our formal proof.
Example. Consider the following information structure
P1 = ff!1; !2g ; f!3; !4g ; :::g = P11; P 2 1; ::: , P2 = ff!1; !2; !3; !4g ; :::g = P21; ::: so that P1 ^ P2 =ff!1; !2; !3; !4g ; :::g .
Suppose agent 1 and 2 have a common neo-additive prior with 2 (0; 1) such that
(f!1g) = ::: = (f!4g) > 0.
and observe that 1 Aj P11 = 1 Aj P12 = opt 1 + 1 opt 1 1 2 (11) with opt 1 = + (1 ) (f!1; !2g) and 2 Aj P21 = opt 2 + 1 opt 2 1 2 (12) with opt 2 = + (1 ) (f!1; !2; !3; !4g) .
Observe that the posterior of each agent is the same in every state belonging to P (! ) 2 P1^ P2 with ! 2 f!1; !2; !3; !4g so that we can stipulate that the agents’posteriors
are common knowledge in every state ! 2 f!1; !2; !3; !4g. Since > 0, the posteriors
(11) and (12) coincide if and only if
opt 1 = opt 2 , (f!1; !2g) = (f!1; !2; !3; !4g) , (P1(! )) = (P2(! )),
which is not the case in this example. Thus, despite identical priors and identical Bayesian learning rules, both agents have di¤erent posterior beliefs which are common knowledge.
Proof of proposition 2. Our proof builds on Aumann’s (1976) original proof for the case of an additive probability measure, i.e., = 0.
Step 1. Aumann (1976): For an additive common prior the agents’ posteriors
must be identical when they are common knowledge at some state of the world.
Suppose to the contrary that there is some ! 2 in which it is common knowledge
that
1(Aj P1(! )) = q1 and 2(Aj P2(! )) = q2
such that q1 6= q2 for some event A 2 B. Then
1 A j P1j = q1; (13)
for all P1j P (! ) whereby P (! ) is the member of P1^ P2 containing ! . Denote by
P1
1; :::; P1n the members of P1 such that
By additivity,
P11 + ::: + (P1n) = (P (! )) (14)
since P1
1; :::; P1n is a partition of P (! ). Also by additivity,
1 Aj P1j = A\ P1j P1j , j = 1; ::; n so that, by (13), P11 + ::: + (P1n) = (A\ P 1 1) q1 + ::: + (A\ P n 1) q1 : Since, by additivity, A\ P11 + ::: + (A\ P1n) = (A\ P (! )) we have P11 + ::: (P1n) = (A\ P (! )) q1 : Thus, by (14), (A\ P (! )) q1 = (P (! )).
An analogous argument for agent 2 results in
(A\ P (! ))
q2
= (P (! ))
implying the desired result q1 = q2.
Step 2. Consider now the case of identical non-additive priors (1), i.e., > 0. Let
A =2 f?; g and suppose both agents are optimistically biased; (there is an analogous
argument for pessimistically biased agents). Then, for ! 2 ,
opt 1 (Aj P1(!)) = opt1;B + 1 opt 1;B (Aj P1(!)) (15) with opt 1;B = + (1 ) (P 1(!)) and opt 2 (Aj P2(!)) = opt2;B + 1 opt 2;B (Aj P2(!)) (16) with opt 2;B = + (1 ) (P (!)):
Assume now that the posteriors (15) and (16) are common knowledge in some state
! 2 . By the argument of step 1, the posteriors must coincide for the special case of
= 0. We therefore have for the additive part of the posteriors (Aj P1(! )) = (Aj P2(! )),
so that the agents’posteriors (15) and (16) are di¤erent if and only if
opt 1;B 6= opt 2;B , (P1(! )) 6= (P2(! )), (P1(! )) 6= (P2(! )).
This proves the proposition.
7
Concluding remarks
In a …rst step, we have developed a model of psychologically biased Bayesian learning whereby we focus on the two benchmark cases of optimistically, resp. pessimistically, biased learning. While our model encompasses the standard model of rational Bayesian learning as a special case, it additionally allows for the possibility that an agent exhibits a “myside bias” in the interpretation of new information.
In a second step, we apply our model of psychologically biased Bayesian learning to the epistemic situation studied in Aumann (1976). Two main results emerge:
1. Even if people receive the same information, they may agree to disagree if their psychologically attitudes about the interpretation of new information are di¤erent. 2. Even if people have the same psychologically attitudes, they may agree to disagree
if they receive di¤erent information.
Both results are in contrast to Aumann’s famous conclusion that agents cannot agree to disagree regardless of whether they receive the same information or not. Our concept of psychologically biased Bayesian learning can therefore o¤er a possible explanation for the existence of ex-post trade in …nancial assets.
Appendix
Proof of observation 1:
Applying the optimistic Bayesian update rule to a neo-additive capacity gives, for A =2 f?; g, (Aj B) = + (1 ) (A\ B) + (1 ) (B) = + (1 ) (B) + (1 ) (B) + (1 ) (B) (Aj B) = optB + 1 optB (Aj B) such that opt B = + (1 ) (B).
Applying the pessimistic Bayesian update rule to a neo-additive capacity gives, for A =2 f?; g, pess(A j B) = (A[ :B)1 ( (:B) :B) = + (1 ) (A[ :B) (1 ) (:B) 1 (1 ) (:B) = (1 ) (A) 1 (1 ) ( (:B)) (1 ) (A\ :B) 1 (1 ) ( (:B)) = (1 ) (A) 1 (1 ) ( (:B)) (1 ) (:B) 1 (1 ) ( (:B)) (Aj :B) = (1 ) (A) 1 (1 ) ( (:B)) (1 ) (:B) 1 (1 ) ( (:B)) (A) (Aj B) (B) (:B) = (1 ) (B) (1 ) + (1 ) (B) (Aj B) = (1 pessB ) (Aj B) such that pess B = (1 ) ( (1 ) + (1 ) (B)).
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